The trade-off among different control loop performances
/Although various advanced control algorithms (e.g. multivariable predictive control) can be successfully implemented in many control problems, simple PID controllers are still in charge of more than 90% of industrial loops and act as base layer for more sophisticated techniques. PID algorithms are widely employed due to their attractive cost to benefit ratio, however their benefits are not always fully exploited and their performance could be improved. In this context it is worth having a clear idea of which kinds of performance can be addressed.
When considering a control task, several different concepts can be defined as control performance such as set-point tracking, disturbance rejection, control effort reduction, and robustness against different operating conditions. These goals cannot be achieved simultaneously and often a ‘trade-off’ between the objectives is required. Improving one objective may mean poor performance in another. Each goal can be translated into design specifications, and specific indices can measure the performance of the PID controller.
Trade-off between tracking and regulation
Set-point tracking can be addressed through the crossover frequency of the loop transfer function. Higher crossover frequency (i.e. the one corresponding to the point where the Nyquist diagram enters in the unit circle) means faster closed loop response. When tuning standard PID controllers, it is hard to achieve good tracking and fast disturbance rejection at the same time. Assuming the control bandwidth is fixed; faster disturbance rejection requires more gain inside the bandwidth, which can only be achieved by increasing the slope near the crossover frequency. As a larger slope means getting closer to the critical (-1, 0i) point, this typically comes at the expense of more overshoot in response to the set-point changes.
For the control room operators who only work based on time domain control performance parameters (e.g. settling time, rise time, and maximum overshoot), these frequency domain parameters are unfamiliar. Therefore, often a more familiar performance indicator Integral of the Absolute Errors (IAE) is considered. A relatively low IAE embeds a relatively fast closed loop response and relatively low oscillatory behavior in the controlled variable.
It is worth stressing that the closed loop transfer function between the set-point and error is different from that of the load disturbance and error; therefore a low IAE in fast tracking task leads to slow-moving behavior with high IAE in the load disturbance rejection; conversely, a quick reaction to the disturbance means high overshoot in response to set-point step change. This is very common when zero-pole cancellation occurs in the closed loop transfer function. The cancellation works for the tracking task but the poles of the process are still there in the transfer function between the load disturbance and the process variable. The typical situations are summarized in the table below:
Trade-off between performance and robustness
Another important trade-off is the one between performance and robustness. The robustness is a measure of how much the controller can tolerate changes in the process transfer function; more specifically, in its gain and in its phase lag.
Also in this case there are some useful parameters which can be used as design criteria for robustness:
- The gain margin is a measure of how much the process gain can change before the closed loop system becomes unstable. Control theory says it is the amount of gain increase or decrease required to make the loop gain unity at the frequency where the phase angle is –180°. More accurately, it is the maximum factor by which the process gain can be multiplied before the closed loop transfer function become unstable.
- The phase margin (Φm) is a measure of how much the process phase can change before the closed loop system become unstable. The process critical phase Φc could be increased because of an additional delay (e.g. due to friction in the valve) or because the process lag decrease (e.g. due to a change in the fluid properties or in the reaction speed); from the control theory, the maximum delay that could be tolerated by the loop is Θ = Φm/ωc (where is the critical pulse corresponding to the point where the Nyquist diagram enters in the unit circle). Furthermore, it’s interesting to remark that when the closed loop system is represented by a second order oscillating system, its damping is somehow proportional to the phase margin.
One interesting single parameter for evaluating the robustness is the so called worst case sensitivity, given by the shorter distance from the Nyquist diagram to the real point -1; this maximum sensitivity is strictly related to the gain and phase margin through some simple inequalities.
Through this single parameter it’s not difficult representing the trade-off between performance and robustness as it happens in the figure (which is referred to a simple specific FOPDT process); of course the trade-off does not continue so much on the right because as soon the Nyquist diagram start approaching to (-1, 0i) the behavior becomes oscillating and IAE increases again.
Additional considerations should be made about the measurement noise filtering and the control effort. The closed loop system acts more or less as a low-pass filter with unit gain and bandwidth equal to [0 ωc]; therefore ωc should be set high enough to allow fast set-point changes and quick reaction to disturbances but not so much to let the measurement noise affect the control effort: for not stressing or saturate the actuator, the controller C(jω) should not have high values for ω > ωc. So it is clear that we have a trade-off again.
Robustness indexes: phase margin, gain margin, worst case sensitivity
Solutions and Conclusions
To achieve an acceptable trade-off between differing control loop performances, some measures can be adopted. One solution is implementing a two Degree of Freedom (2DoF) controller where the ordinary PID parameters can be tuned for good disturbance rejection, but where an additional set-point offsets the overshoot in the set-point tracker. Where the 2DoF algorithm is not available, set-point changes should be ramp instead of step.
Another solution is to employ different sets of PID parameters depending on the control problem or the process situation. The default PID controller should be for providing an effective reaction to unpredictable disturbances; then a different PID controller can be recalled any time a set-point is changed by the operator. When it is known that the plant is going to work under unusual or transient conditions, a set of robustness-oriented PID parameters can be copied in the controller memory.
The PID algorithm is one of the simplest solutions to control problems but it cannot provide effective results for different kinds of performance and objectives. Focusing on the more relevant tasks and selecting the set of parameters more appropriate for it; alternatively more complex techniques can be implemented in order to make the PID controller effective in different operating conditions.
